# Gerolamo Cardano

Italian Renaissance mathematician, physician, astrologer

Gerolamo Cardano (September 24, 1501September 21, 1576) was an Italian Renaissance mathematician, physician, astrologer and gambler, applying probability calculus to games of chance.

Gerolamo Cardano

## Quotes

• The greatest advantage in gambling lies in not playing at all.
• Gerolamo Cardano (around 1560). Liber de ludo aleae.

### Cardanus Comforte (1574)

Thomas Bedingfield's translation of De Consolatione (1542)

• Better it is to have the worst, than none at all. for example we see, that houses are nedefull, such as can not possese & stately pallaces of stone, do persuade themselves to dwell in houses of timber and clap, and wanting them, are contented to inhabite the simple cotage, yea rather than not to be housed at all refuse not the pore cabbon, and most beggerly cave. So necessarie is this gift of consolacion, as there livith no man, but that hathe cause to embrace it. for in these things better is it to have any than none at al.
• And wel we see ther is none alive that in every respect may be accompted happie, yea though mortall men were free from all calamities, yet the torments & feare of death should stil attend them But b:sides them, behold, what, and how manye evilles there bee, that unlesse the cloude of error bee removed, impossible it is to see the truth, or receive allay of our earthly woes.
• So shall we voyd of all craft and sail, with true reason declare how much each man erreth in life, judgement, opinion, and will. Some things there are that so wel do prove themselves, as besides nature nede no profe at all.
• Among other myseries what I pray you tá be greater than whē a man riseth frō bed in the morning, to be incertaine of his returne to rest againe. or being in bed, whether his life shall continue tyll he ryse. besydes that, what labour, what hazard & care, are men constrained to abyde with these our brittle bodies, our feeble force, and incertayne lyfe: so as no nacion I thinke a man better or more fitlye named than the Spaniard, who in their language do terme a man shadow. And sure ther is nothing to be found of lesse assurance or soner passed than the lyfe of man, no... may more rightly be resembled to a shadow.

### The Book of My Life (1930)

Jean Stoner translation of De Vita Propria Liber (1654)

• From these beginnings, as it were, have issued bitterness, contentious obstinancy, lack of amenity, hasty judgement, anger, and an intense desire for revenge—to say nothing of headstrong will; that which many damn, by word at least, was my delight.
• I am cold of heart, warm of brain, and given to never-ending meditation; I ponder over ideas, many and weighty, and even over things which can never come to pass.
• I am able to admit two distinct trains of thought to my mind at the same time.
• I have accustomed my features always to assume an expression quite contrary to my feelings; thus I am able to feign outwardly, yet within know nothing of dissumulation. This habit is easy if compared to the practice of hoping for nothing, which I have bent my efforts toward acquiring for fifteen successive years, and have at last succeeded.
• My personal affairs are not as highly esteemed as men commonly value their own interests—vain, empty affairs like those great clouds seen in the wake of the sunset which are meaningless and soon pass away.
• This I recognize as unique and outstanding among my faults—the habit...of preferring to say above all things what I know to be displeasing to the ears of my hearers. ...I keep it up wilfully, in no way ignorant of how many enemies it makes for me. ...Yet I avoid this practice in the presence of my benefactors and of my superiors. It is enough not to fawn upon these, or at least not to flatter them.
• What if one should address a word to the kings of the earth and say, "Not one of you but eats lice, flies, bugs, worms, fleas—nay the very filth of your servants! With what an attitude would they listen to such statements, though they be truths? What is this complacency then but an ignoring of conditions, a pretense of not being aware of what we know exists, or a will to set aside a fact by force? And so it is with everything else foul, vain, confused and untrue in our lives.
• I have not lost my faith; and this I must attribute more to a miracle than to my own wisdom; more to Divine Providence than to my own virtue. Steadfastly, in fact from my earliest childhood, I have made this my prayer, "Lord God... grant me long life, and wisdom, and health of mind and body."
• My father, in my earliest childhood, taught me the rudiments of arithmetic, and about that time made me acquainted with the arcana; whence he had come by this learning I know not. This was about my ninth year. Shortly after, he instructed me in the elements of the astronomy of Arabia, meanwhile trying to instill in me some system of theory for memorizing, for I had been poorly endowed with the ability to remember. After I was twelve years old he taught me the first six books of Euclid, but in such a manner that he expended no effort on such parts as I was able to understand by myself.
This is the knowledge I was able to acquire and learn without any elementary schooling...

### The Great Rules of Algebra (1968)

T. Richard Witmer translation of Artis Magnae, sive de Regulis Algebraicis (1545)

• Since this art surpasses all human subtelty and the perspecuity of mortal talent and is truly a celestial gift and a very clear test of the capacity of man's minds, whoever applies himself to it will believe that there is nothing that he cannot understand.
• Although a long series of rules might be added and a long discourse given about them, we conclude our detailed consideration with the cubic, others being merely mentioned, even if generally, in passing. For as positio refers to a line, quadratum to the surface, and cubum to a solid body, it would be very foolish for us to go beyond this point. Nature does no permit it.

• You troubled mindes with tormentes loste
that sighes and sobs consumes:
(Who breathes and puffes from burning breast,
both smothring smoke and fumes.)
Come reade this booke that freelye bringes, a boxe of balme full swete,
An oyle to noynt the brused partes, of everye heavye spirete.
...The lame whose lack of legges is death, unto a loftye mynde,
Wyll kiss his crotche and creepe on knees,Cardanus workes to fynde.
• Thomas Churchyarde, in behalfe of the Booke, Cardanus Comforte (1574) Thomas Bedingfield's translation of De Consolatione (1542)
• Every medieval and Renaissance court had a royal astrologer who advised the duke or prince he served. ...Men such as Roger Bacon, who even in the thirteenth century was a clear and outspoken champion of the experimental method in science, and Jerome Cardan, one of the foremost mathematicians and physicians of the sixteenth century, subscribed to astrology.

### Introduction to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries (1866)

Gerolamo Cardano
• Jerome Cardan is... the founder of the higher algebra; for, whatever he may have borrowed from others, we derive the science from his Ars Magna, published in 1545. It contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. The original inventor appears to have been Scipio Ferreo, who, about 1505, by some unknown process, discovered the solution of a single case; that of x3 + px = q. Ferreo imparted the secret to one Fiore, or Floridus, who challenged Tartaglia to a public trial of skill, not unusual in that age. Before he heard of this, Tartaglia, as he assures us himself, had found out the solution of two other forms of cubic equation; x3 + px2 = q, and x3 - px2 = q. When the day of trial arrived, Tartaglia was able, not only to solve the problems offered by Fiore, but to baffle him entirely by others which resulted in the forms of equation, the solution of which had been discovered by himself. This was in 1535; and, four years afterwards, Cardan obtained the secret from Tartaglia under an oath of secrecy. In his Ars Magna, he did not hesitate to violate this engagement; and, though he gave Tartaglia the credit of the discovery, revealed the process to the world.
• Anticipations of Cardan are more truly wonderful when we consider that the symbolical language of algebra, that powerful instrument not only expediting the processes of thought, but in suggesting general truths to the mind, was nearly unknown in his age. Diophantus, Fra Luca, and Cardan make use occasionally of letters to express indefinite quantities besides the res or cosa, sometimes written shortly, for the assumed unknown number of an equation. But letters were not yet substituted for known quantities. Michael Stifel, in his Arithmetics Integra, Nuremberg, 1544, is said to have first used the signs + and -, and numeral exponents of powers. It is very singular that discoveries of the greatest convenience, and apparently, not above the ingenuity of a village schoolmaster, should have been overlooked by men of extraordinary acuteness like Tartaglia, Cardan, and Ferrari; and hardly less so, that by dint of this acuteness they dispensed with the aid of these contrivances, in which we suppose that so much of the utility of algebraic expression consists.

### A Short Account of the History of Mathematics (1888)

W. W. Rouse Ball Note: later editions include 1893, 1901 & 1905

• His career is an account of the most extraordinary and inconsistent acts. A gambler, if not a murderer, he was an ardent student of science, solving problems which had long baffled all investigation; at one time in his life he was devoted to intrigues which were a scandal even in the sixteenth century, at another he did nothing but rave on astrology, and yet at another he declared that philosophy was the only subject worthy of a man's attention. His was the genius that was closely allied to madness.
• After spending a year or so in France, Scotland, and England, he returned to Milan as professor of science, and shortly afterward was elected to a chair at Pavia. Here he divided his time between debauchery, astrology, and mechanics. His two sons were as wicked and passionate as himself: the elder was in 1560 executed for poisoning his wife, and about the same time Cardan in a fit of rage cut off the ears of the younger who committed some offence; for this scandelous outrage he suffered no punishment, as Pope Gregory XIII granted him protection.
• In 1570 he was imprisoned for heresy on account of his having published the horoscope of Christ, and when released he found himself... generally detested...
• Cardan was the most distinguished astrologer of his time, and when he settled in Rome he received a pension in order to secure his services as astrologer to the papal court. This proved fatal to him for, having foretold that he should die on a particular date, he felt obliged to commit suicide in order to keep up his reputation—so at least the story runs.
• The Ars Magna is a great advance on any algebra previously published. Hitherto algebraists had confined their attention to those roots of equations which were positive. Cardan discussed negative and even complex roots, and proved that the latter would always occur in pairs, though he declined to commit himself to any explanation as to the meaning of these "sophistic" quantities which he said were ingenious though useless.
• Most of his analysis of cubic equations seems to have been original; he shewed that if the three roots were real, Tartaglia's solution gave them in a form which involved imaginary quantities. Except for the somewhat similar researches of Bombelli a few years later, the theory of imaginary quantities received little further attention from mathematicians until John Bernoulli and Euler took up the matter after the lapse of nearly two centuries. Gauss first put the subject on a systematic and scientific basis, introduced the notation of complex variables, and used the symbol i, which had been introduced by Euler in 1777, to denote the square root of (-1): the modern theory is chiefly based on his researches.

### History of Mathematics (1925) Vol. 2

• The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). This was devoted primarily to the solution of algebraic equations. It contained the solution of the cubic and biquadratic equations, made use of complex numbers, and in general may be said to have been the first step toward modern algebra.
• [Zuanne de Tonini] da Coi... impuned Tartaglia to publish his method, but the latter declined to do so. In 1539 Cardan wrote to Tartaglia, and a meeting was arranged at which, Tartaglia says, having pledged Cardan to secrecy, he revealed the method in cryptic verse and later with a full explanation. Cardan admits that he received the solution from Tartaglia, but... without any explanation. At any rate, the two cubics ${\displaystyle x^{3}+ax^{2}=c}$  and ${\displaystyle x^{3}+bx=c}$  could now be solved. The reduction of the general cubic ${\displaystyle x^{3}+ax^{2}+bx=c}$  to the second of these forms does not seem to have been considered by Tartaglia at the time of the controversy. When Cardan published his Ars Magna however, he transformed the types ${\displaystyle x^{3}=ax^{2}+c}$  and ${\displaystyle x^{3}+ax^{2}=c}$  by substituting ${\displaystyle x=y+{\frac {1}{3}}a}$  and ${\displaystyle x=y-{\frac {1}{3}}a}$  respectively, and transformed the type ${\displaystyle x^{3}+c=ax^{2}}$  by the substitution ${\displaystyle x={\sqrt[{3}]{c^{2}/y}},}$  thus freeing the equations of the term ${\displaystyle x^{2}}$ . This completed the general solution, and he applied the method to the complete cubic in his later problems.
• Cardan's originality in the matter seems to have been shown chiefly in four respects. First, he reduced the general equation to the type ${\displaystyle x^{3}+bx=c}$ ; second, in a letter written August 4, 1539, he discussed the question of the irreducible case; third, he had the idea of the number of roots to be expected in the cubic; and, fourth, he made a beginning in the theory of symmetric functions. ...With respect to the irreducible case... we have the cube root of a complex number, thus reaching an expression that is irreducible even though all three values of x turn out to be real. With respect to the number of roots to be expected in the cubic... before this time only two roots were ever found, negative roots being rejected. As to the question of symmetric functions, he stated that the sum of the roots is minus the coefficient of x2
• He states that the root of ${\displaystyle x^{3}+6x=20}$  is
${\displaystyle x={\sqrt[{3}]{{\sqrt {108}}+10}}-{\sqrt[{3}]{{\sqrt {108}}-10}}.}$
• He... gave thirteen forms of the cubic which have positive roots, these having already been given by Omar Kayyam.
• The problem of the biquadratic equation was laid prominently before Italian mathematicians by Zuanne de Tonini da Coi, who in 1540 proposed the problem, "Divide 10 parts into three parts such that they shall be continued in proportion and that the product of the first two shall be 6." He gave this to Cardan with the statement that it could not be solved, but Cardan denied the assertion, although himself unable to solve it. He gave it to Ferrari, his pupil, and the latter, although then a mere youth, succeeded where the master had failed. ...This method soon became known to algebraists through Cardan's Ars Magna, and in 1567 we find it used by Nicolas Petri [of Deventer].
• The law which asserts that the equation X = 0, complete or incomplete, can have no more real positive roots than it has changes of sign, and no more real negative roots than it has permanences of sign, was apparently known to Cardan; but a satisfactory statement is possibly due to Harriot (died 1621) and certainly to Descartes.
• The application of the theory [of probability] to mortality tables in any large way may be said to have started with John Graunt... The first tables of great importance, however, were those of Edmund Halley... however... Cardan seems to have been the first to have been the first to consider the problem in a printed work, although his treatment is very fanciful. He gives a brief table in his proposition "Spatium vitae naturalis per spatium vitae fortuitum declarare," this appearing in the De Proportionibus Libri V...

### Cardano the Gambling Scholar (1953)

• Cardano's entertaining books on science and curiosities were among the best read and most pirated works in the sixteenth century. ...his work on the "Great Art" has been characterized as the first that goes decisively beyond the attainments of classical Greek mathematics.
• Most important for the history of science is the fact that Liber de Ludo Aleae, "The Book of Games of Chance," contains the first study of the principles of probability. ...it would seem much more just to date the beginnings of probability theory from Cardano's treatise rather than the customary reckoning from Pascal's discussions with his friend de Méré and the ensuing correspondence with Fermat... at least a century after Cardano...
• Cardano was a man of universal interests, and much of his ability must have been inherited from his father, Fazio Cardano... a lawyer... but also deeply steeped in the medical sciences, mathematics, and all kinds of occult lore... he had... a high reputation as a scholar in his native town; even Leonardo da Vinci notes several times that he consulted Messer Fazio on geometric questions... he was appointed as a public lecturer in geometry.
• When Cardano's Consolation or Comforte was translated into English in 1573... one of the readers is known to have been William Shakespeare. ...Hamlet's thoughts on death and slumber are believed to have been inspired by... passages in Comforte...
• A great number of writers on the history of medicine have indicated important observations and suggestions which made their intitial appearance with Cardano.